Let
${a}_{n},\phantom{\rule{0.2em}{0ex}}{b}_{n}\ge 0$ for all
$n\ge 1.$
If
$\underset{n\to \infty}{\text{lim}}{a}_{n}\text{/}{b}_{n}=L\ne 0,$ then
$\sum}_{n=1}^{\infty}{a}_{n$ and
$\sum}_{n=1}^{\infty}{b}_{n$ both converge or both diverge.
If
$\underset{n\to \infty}{\text{lim}}{a}_{n}\text{/}{b}_{n}=0$ and
$\sum}_{n=1}^{\infty}{b}_{n$ converges, then
$\sum}_{n=1}^{\infty}{a}_{n$ converges.
If
$\underset{n\to \infty}{\text{lim}}{a}_{n}\text{/}{b}_{n}=\infty $ and
$\sum}_{n=1}^{\infty}{b}_{n$ diverges, then
$\sum}_{n=1}^{\infty}{a}_{n$ diverges.
Note that if
${a}_{n}\text{/}{b}_{n}\to 0$ and
$\sum}_{n=1}^{\infty}{b}_{n$ diverges, the limit comparison test gives no information. Similarly, if
${a}_{n}\text{/}{b}_{n}\to \infty $ and
$\sum}_{n=1}^{\infty}{b}_{n$ converges, the test also provides no information. For example, consider the two series
$\sum _{n=1}^{\infty}1}\text{/}\sqrt{n$ and
$\sum _{n=1}^{\infty}1}\text{/}{n}^{2}.$ These series are both
p -series with
$p=1\text{/}2$ and
$p=2,$ respectively. Since
$p=1\text{/}2>1,$ the series
$\sum _{n=1}^{\infty}1}\text{/}\sqrt{n$ diverges. On the other hand, since
$p=2<1,$ the series
$\sum _{n=1}^{\infty}1}\text{/}{n}^{2$ converges. However, suppose we attempted to apply the limit comparison test, using the convergent
$p-\text{series}$$\sum _{n=1}^{\infty}1}\text{/}{n}^{3$ as our comparison series. First, we see that
Therefore, if
${a}_{n}\text{/}{b}_{n}\to \infty $ when
$\sum _{n=1}^{\infty}{b}_{n}$ converges, we do not gain any information on the convergence or divergence of
$\sum _{n=1}^{\infty}{a}_{n}}.$
Using the limit comparison test
For each of the following series, use the limit comparison test to determine whether the series converges or diverges. If the test does not apply, say so.
In order to evaluate
$\underset{n\to \infty}{\text{lim}}\text{ln}\phantom{\rule{0.1em}{0ex}}n\text{/}n,$ evaluate the limit as
$x\to \infty $ of the real-valued function
$\text{ln}(x)\text{/}x.$ These two limits are equal, and making this change allows us to use L’Hôpital’s rule. We obtain
Since the limit is
$0$ but
$\sum}_{n=1}^{\infty}\frac{1}{n$ diverges, the limit comparison test does not provide any information.
Compare with
$\sum}_{n=1}^{\infty}\frac{1}{{n}^{2}$ instead. In this case,
Since the limit is
$\infty $ but
$\sum}_{n=1}^{\infty}\frac{1}{{n}^{2}$ converges, the test still does not provide any information.
So now we try a series between the two we already tried. Choosing the series
$\sum}_{n=1}^{\infty}\frac{1}{{n}^{3\text{/}2}},$ we see that
As above, in order to evaluate
$\underset{n\to \infty}{\text{lim}}\text{ln}\phantom{\rule{0.1em}{0ex}}n\text{/}\sqrt{n},$ evaluate the limit as
$x\to \infty $ of the real-valued function
$\text{ln}\phantom{\rule{0.1em}{0ex}}x\text{/}\sqrt{x}.$ Using L’Hôpital’s rule,
Since the limit is
$0$ and
$\sum}_{n=1}^{\infty}\frac{1}{{n}^{3\text{/}2}$ converges, we can conclude that
$\sum}_{n=1}^{\infty}\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}n}{{n}^{2}$ converges.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?